metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1F5, C40⋊1C4, D5.1D8, D10.9D4, D5.1Q16, Dic5.3Q8, C5⋊(C2.D8), C5⋊2C8⋊6C4, C4⋊F5.4C2, C4.9(C2×F5), C20.9(C2×C4), (C8×D5).3C2, C2.5(C4⋊F5), C10.2(C4⋊C4), (C4×D5).26C22, SmallGroup(160,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5.D8
G = < a,b,c,d | a5=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Character table of D5.D8
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
size | 1 | 1 | 5 | 5 | 2 | 10 | 20 | 20 | 20 | 20 | 4 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
ρ12 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ15 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
ρ17 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4⋊F5 |
ρ18 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4⋊F5 |
ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | complex faithful |
ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | complex faithful |
ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | complex faithful |
(1 26 37 18 10)(2 27 38 19 11)(3 28 39 20 12)(4 29 40 21 13)(5 30 33 22 14)(6 31 34 23 15)(7 32 35 24 16)(8 25 36 17 9)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)
(1 8)(2 7)(3 6)(4 5)(9 18 25 37)(10 17 26 36)(11 24 27 35)(12 23 28 34)(13 22 29 33)(14 21 30 40)(15 20 31 39)(16 19 32 38)
G:=sub<Sym(40)| (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38)>;
G:=Group( (1,26,37,18,10)(2,27,38,19,11)(3,28,39,20,12)(4,29,40,21,13)(5,30,33,22,14)(6,31,34,23,15)(7,32,35,24,16)(8,25,36,17,9), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40), (1,8)(2,7)(3,6)(4,5)(9,18,25,37)(10,17,26,36)(11,24,27,35)(12,23,28,34)(13,22,29,33)(14,21,30,40)(15,20,31,39)(16,19,32,38) );
G=PermutationGroup([(1,26,37,18,10),(2,27,38,19,11),(3,28,39,20,12),(4,29,40,21,13),(5,30,33,22,14),(6,31,34,23,15),(7,32,35,24,16),(8,25,36,17,9)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40)], [(1,8),(2,7),(3,6),(4,5),(9,18,25,37),(10,17,26,36),(11,24,27,35),(12,23,28,34),(13,22,29,33),(14,21,30,40),(15,20,31,39),(16,19,32,38)])
D5.D8 is a maximal subgroup of
C80⋊2C4 C80⋊3C4 D5.D16 D5.Q32 (C2×C8)⋊6F5 M4(2)⋊1F5 D8×F5 SD16⋊F5 Q16×F5 Dic5.4Dic6 D5.D24
D5.D8 is a maximal quotient of
C80⋊2C4 C80⋊3C4 C16.F5 C80.2C4 C40⋊1C8 Dic5.13D8 D10.10D8 Dic5.4Dic6 D5.D24
Matrix representation of D5.D8 ►in GL4(𝔽7) generated by
3 | 1 | 3 | 3 |
0 | 1 | 6 | 3 |
4 | 4 | 2 | 2 |
1 | 5 | 6 | 0 |
6 | 0 | 4 | 2 |
0 | 6 | 1 | 4 |
0 | 3 | 3 | 1 |
0 | 1 | 3 | 6 |
3 | 3 | 0 | 2 |
1 | 3 | 1 | 1 |
2 | 2 | 6 | 3 |
3 | 4 | 2 | 3 |
4 | 1 | 3 | 3 |
6 | 2 | 6 | 6 |
1 | 2 | 1 | 2 |
0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(7))| [3,0,4,1,1,1,4,5,3,6,2,6,3,3,2,0],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[4,6,1,0,1,2,2,2,3,6,1,1,3,6,2,0] >;
D5.D8 in GAP, Magma, Sage, TeX
D_5.D_8
% in TeX
G:=Group("D5.D8");
// GroupNames label
G:=SmallGroup(160,69);
// by ID
G=gap.SmallGroup(160,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D5.D8 in TeX
Character table of D5.D8 in TeX